Why can't we take a limiting relative frequency over nested spheres in relativistic spacetime? The Measure Problem of eternal inflation cosmology is to determine the correct probability measure over the theoretically infinitely many individuals (or civilizations or worlds) for self-locating belief.  Ultimately we want to know what credence we should we assign to being in  a world with property P, according to a given model, so that we can judge how much the observation of P or not-P confirms or disconfirms the model.
A seemingly natural answer is just to take a finite sphere in spacetime, estimate the portion of observers in it where P holds out of all individuals in the sphere, then repeatedly expand the sphere and take the frequency of individuals observing P in the limit as the sphere grows without bound.
But at least one discussion of this problem asserts that, "in relativistic spacetime, the proposal to assign probabilities by taking limiting relative frequencies in sequences of nested spheres doesn’t even make sense" (Arntzenius and Dorr 2017).  Why not?
[EDIT 3 Oct 2020 - Earlier in the paper they write, "in relativistic space-times there is no useful notion of a ‘four-dimensional sphere’—the closest analogues of spheres are regions bounded by hyperboloids, but these regions will in general contain infinite numbers of observers and hence be useless for the purpose of taking limits."
So they seem to be arguing that the procedure doesn't even make sense in Special Relativity or a flat Minkowski spacetime.]
Never mind all the other issues about the reference class problem, what counts as an observer, etc.  I just want to know why taking limiting relative frequencies in sequences of nested spheres doesn’t make sense in relativistic spacetime.
[EDIT - Assume a flat Minkowski spacetime to begin with. "Spheres" in the Minkowski metric aren't finite, so that won't work.  And spheres in one observer's coordinates won't be spheres for another.  But can't we at least define nested bounded sets that grow without bound?]
Update:  Detailed reference
Arntzenius, Frank, and Cian Dorr. 2017. “Self-Locating Priors and Cosmological Measures”. In The Philosophy of Cosmology, ed. by Khalil Chamcham et al., 396–428. Cambridge: Cambridge University Press.
http://philsci-archive.pitt.edu/11864/
 A: I think commentator @probably_someone has the right answer.  Nested spheres in relativistic spacetime don't make sense because a sphere in one reference frame is not a sphere in others.  A natural idea, which Dorr and Arntzenius hint at, is to take spheres in the invariant Minkowski metric (the spacetime interval), but these will generally be hyperboloids of infinite volume and therefore not helpful for taking finite frequencies.
However, I think D&A have made a dialectical error by resting too much on the notion of a sphere per se.  If we just want to take a limiting relative frequency, it won't matter whether we take it over a sequence of spheres, ellipsoids, or any other bounded regions, provided the choice of regions is not biased towards the actual distribution of properties.  Concentric spheres in one reference frame will still be nested bounded regions in every other reference frame.
The relevant question, then, is this:  Is there a legitimate worry that we will get different frequencies depending on which reference frame we use to define our spheres?  Such variance would require a peculiar distribution of properties that is biased towards certain reference frames, and to argue for that would require considerations beyond mere relativity.
So in sum, frame-independent spheres are not well defined in any relativistic spacetime, but relativity on its own does not rule out taking a limiting relative frequency over bounded regions of spacetime.  If there is any obstacle to such a procedure, it must come from further considerations:  either (i) a peculiar frame-biased distribution of properties, (ii) considerations of mass and curvature in General Relativity, or (iii) other aspects of Eternal Inflation.  I would love to know what exactly the main obstacle is, but that is another question.
